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1919] RANDOM FLIGHTS IN ONE, TWO, OB. THREE DIMENSIONS 619 In proceeding to the general value of n, we may conveniently follow the analogy of the two-dimensional investigation of Kluyver, for which purpose we require a function that shall be unity when s < r, and zero when s > r. Such a function is 2 fm 7 sin sx sin rx — rw cos rx __ I riff_____..____________ ___ I \JJMJ ' • 7T J o SX X .(51) for it may be written 7T6'J0 , fsmrx\ 2 f smrsc , sin seed-------= — -------cos sxdx \ rcc ) TrJo oo 1 T00 sin (s + r)a} — $m(s — r}x7 ., A = — I-------------J-------------^-------— doc —I or 0, IT Jo &' according as s is less or greater than r. In like manner for a second lemma, corresponding with (25), we may reason again from the triangle GFE (Fig. 1). J0(g) is replaced by Bmg/g, a potential function symmetrical in three dimensions about E and satisfying everywhere V2+ 1 = 0. It may be expanded about G in Legeudre's series* sine 'sin e cos e Lo \ e JrV e2 being written for cos (?, and accordingly 1 sinV(e2+/2 ^ ~777^rT7a" W. sine When E and jP are interchanged, the same integral is seen to be proportional to sin///, and may therefore be equated to . , sin e sin/ ^0 — -7-, where A0' is now an absolute constant, whose value is determined to be unity by putting e, or / equal to zero. We may therefore write 1 f+ij sin V(*+/'-2«<//0_ sine sin/ 2 J -i ^ "7^ +« - 2fl~ " "7" As in the case of two dimensions, similar reasoning shows that , cos V(e3 + /2 — 2e/^t) _ sin e cos/ "" provided e </. With appropriate changes, we may now follow Kluyver's argument for two dimensions. The same diagram (Fig. 2) will serve, only the successive triangles are no longer limited to lie in one plane. Instead of the angles 0, we have now to deal with their cosines, of which all values are to be regarded Theory of Sound, § 330.ar.